The AIR Professional FileFall 2019 Volume

Supporting quality data and decisions for higher education.

Association for Institutional Research© Copyright 2019, Association for Institutional Research

This volume of the AIR Professional File illustrates

creative approaches to data analysis in two familiar

areas of institutional research – class size and

enrollment projection.

In Perception Isn’t Everything: The Reality of Class

Size, Umbricht and Stange challenge traditional

measures of class size found in institutional fact

books, used by ranking publications, and computed

for the Common Data Set. What if, instead, we

evaluated class size through the lens of the student

experience? The authors explain how to compute a

student-centric metric of class size and offer advice

for the effective use of this more sophisticated and

nuanced measure.

Who among us hasn’t been pressed to find a magic

formula to accurately forecasts student enrollment?

In Enrollment Projection Using Markov Chains, Gandy,

Crosby, Luna, Kasper and Kendrick show how an

underutilized forecasting tool can be enlisted to

better understand and predict student flow, even in

subgroups of students. With extensive description

of the methodology and presentation of results, the

authors provide a blueprint for replication at your

institution.

Do you have a creative approach to data analysis?

Your colleagues are waiting for you to share it

through the AIR Professional File!

Sincerely,

Sharron Ronco

Editors

Sharron Ronco

Coordinating Editor

Marquette University (retired)

Stephan C. Cooley

Managing Editor

Association for Institutional Research

ISSN 2155-7535

LETTER FROM THE EDITOR

Perception Isn’t Everything: The Reality of Class Size

Article 146 Authors: Mark Umbricht and Kevin Stange

Enrollment Projection Using Markov Chains: Detecting Leaky Pipes and the Bulge in the Boa

Article 147 Authors: Rex Gandy, Lynne Crosby, Andrew Luna, Daniel Kasper, and Sherry Kendrick

Thank You

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Table of Contents

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4Fall 2019 Volume

Mark Umbricht and Kevin Stange

About the Authors

The authors are with the University of Michigan.

Kevin Stange is also with the National Bureau of

Economic Research.

Acknowledgments

We would like to thank Paul Courant, Ben Koester,

Steve Lonn, Tim McCay, and numerous participants

in seminars at the University of Michigan for helpful

comments. This research was conducted as part

of the University of Michigan Institutional Learning

Analytics Committee. The authors are solely

responsible for the conclusions and findings.

Abstract

Every term, institutions of higher education must make decisions about the class size for each class they offer,

which can have implications for student outcomes, satisfaction, and cost. These decisions must be made

within the current higher education landscape of tightening budgets and calls for increased productivity.

Beyond institution decision making, prospective students and their families may use class size as one factor

in deciding whether an institution might be a good fit for them. The current measure of class size found in

university fact books, and subsequently sent to numerous ranking groups such as U.S. News & World Report

(hereafter U.S. News), is an inadequate gauge of the student experience in the classroom, as measured

by the percent of time students spend in classes of varying sizes. The current measure does not weight

for enrollment, credits, or multiple components of a class, which results in a misleading representation of

the student experience of class size. This paper will discuss these issues in depth, explain how class size

varies across institutions, and offer recommendations on how to reweight class size in the Common Data

Set to accurately describe it from the student’s perspective. Institutions could use this new metric to better

understand class size, and subsequently to understand the student experience and cost of a class, while

prospective students and their families could use the metric to gain a clearer picture of the class sizes they

are likely to experience on campus.

Keywords: class size, productivity, student outcomes

Perception Isn’t Everything: The Reality of Class Size

The AIR Professional File, Fall 2019

Article 146

https://doi.org/10.34315/apf1462019

© Copyright 2019, Association for Institutional Research

5Fall 2019 Volume

INTRODUCTIONOrganizing class delivery is a key operational

decision for institutions of higher education. Each

term these institutions must decide how many

students will be taught within each section given

the classes they offer, the faculty and instructors

they have available to teach, and the confines of

physical spaces they have on campus. Within these

constraints, institutions must decide how to deliver

classes. Consider a popular class taken by nearly

every first-year student: Should this class be taught

as one large lecture by a famous professor, many

small sections taught by graduate students, or a

combination of the two? Should small classes target

freshmen who are acclimating to college or seniors

as they specialize in their field?

Institutions, particularly public institutions, make

these decisions within the current context of

increased accountability and decreased resources.

Traditional wisdom argues that smaller classes

increase engagement, facilitate student-faculty

interactions, and improve student success. The

opportunity to learn from prominent scholars in the

field is also considered a strength of undergraduate

education at research universities. However, smaller

classes and senior faculty are costlier and their use

comes at the expense of other ways of enriching

and supporting the undergraduate student

experience. As Courant and Turner (forthcoming)

argue, institutions have an interest in providing

curriculum efficiently, meaning they must strike a

balance between quality, costs, and tuition revenue.

If an institution or department has an influx of

students, it must decide whether it will increase

the size of its faculty or the class size of its courses.

Therefore, decisions about class size have first-order

influence on student success and institutional costs.

From the student perspective, class size could be

influential in the college choice process, with some

students seeking intimate class settings with small

class sizes, and others preferring to blend in to a

large classroom. Students and their families rely

on institutional websites and rankings, such as

Princeton Review or U.S. News, and other publicly

available data for information about class size. These

data are typically drawn from the Common Data Set

(CDS), which is a collaborative effort among data

providers and publishers to improve the quality and

accuracy of information provided to prospective

students, and to reduce the reporting burden on

data providers (CDS Initiative, 2018). While it is

helpful to have a measure that can be reported

across multiple campuses, the class size metric

used by the CDS is measured at the classroom level

rather than at the student level. This difference in

measurement leads to a disconnect between the

metric and the phenomena it is trying to describe, as

described in the following example.

Imagine a high school student researching her

nearby public, research university as a prospective

student. She sees that, according to U.S. News

in 2018, only 17% of classes have more than 50

students, and 57% of classes have fewer than 20

students. The student thinks, What luck! She thinks

she can attend a high-quality research institution

while spending most of her time in small classes.

After graduation from that college, the same

student looks back and sees that she spent more

than 41% of her time in classes with more than

50 students, and only 20% of her time in classes

with fewer than 20 students. These differences in

the perception versus reality are not exaggerated,

but rather are many students’ average experience.

This paper will show that the measure of class size

calculated for the CDS, and subsequently used by

many other sources, does not provide an accurate

approximation of the true class size experienced

6Fall 2019 Volume

by students at the University of Michigan (U-M), a

large, research university in the Midwest. By the

term “student experience,” we mean the percent

of time or credits spent in classes of varying sizes.

The results of this case study could be replicated at

any institution, with varying degrees of departure

from the true student experience depending on the

institution type and size. Specifically, this paper will

argue for a new class size metric to be used in the

CDS and will address the following questions:

Framing Questions

1| How does the standard definition of class size

vary from the student experience?

2| How does a student-centric version of class size

vary across an institution?

3| How can institutional researchers practically

recalculate class size to better approximate

the student experience without significantly

increasing the burden of data providers?

Importance of the Topic and Literature Review

This topic is important for students, institutions,

and the field of institutional research. From the

student perspective, students and those assisting

in their decisions need accurate and meaningful

information to make the best decision about which

college to attend. A small number of studies have

shown that class size is an important factor for

students as they select an institution (Drewes &

Michael, 2006; Espinoza, Bradshaw, & Hausman,

2002). This makes sense since lower class size is

perceived to be linked to gains in student outcomes.

Literature in the secondary setting is clear that lower

class size is associated with gains across multiple

areas, including test scores, noncognitive skills,

college enrollment, and other outcomes (Angrist &

Lavy, 1999; Chetty et al., 2010; Dee & West, 2011;

Dynarski, Hyman, & Schanzenbach, 2013; Hoxby,

2000; Krueger, 1999). However, in higher education

the relationship between class size and outcomes

is not well established, with studies finding either

negligible association (Bettinger, Doss, Loeb, Rogers,

& Taylor, 2017; Lande, Wright, & Bartholomew,

2016; Stange & Umbricht, 2018; Wright, Bergom,

& Lande, 2015) or a negative relationship between

class size and outcomes (Bettinger & Long, 2018; De

Giorgi, Pellizzari, & Woolston, 2012; Kokkelenberg,

Dillon, & Christy, 2008). Institutions that gain a more

accurate and more nuanced version of class size

from the student experience perspective could

aid prospective students in their decision-making

process.

Class size is also important to institutions

for planning purposes. Courant and Turner

(forthcoming) argue that institutions must strike a

balance between quality, costs, and tuition revenue.

In recent years, institutions have been asked to

cut back and do more with fewer resources, which

would imply that increasing class size would be an

appropriate strategy. In fact, class size is one of

the most important drivers of instructional costs

(Hemelt, Stange, Furquim, Simon, & Sawyer, 2018).

However, lower class size is perceived to lead to

better student outcomes and is subsequently tied to

rankings such as those at U.S. News. This common

perception pulls institutions to keep class size lower,

putting institutions in a situation where a logical

solution is to hire cheaper instructors, such as

noncontingent faculty. The ultimate decision on how

to strike this balance is not traditionally made at the

institution level, but rather at the department level.

Cross and Goldenberg (2009) found that the number

of noncontingent faculty at elite research institutions

rose significantly in the 1990s, which was due to

7Fall 2019 Volume

micro- (department-)level decisions. Departments

(or colleges) that are particularly concerned with

the quality (or perceived quality) of small class sizes

would find it difficult to adequately assess how

much time their students spend in classes of a given

size with the current metric, which is at the class

level. A new student experience version of class

size would allow departments to compare the class

size experience across multiple majors or between

departments, which could assist in balancing

the class size constraints for long-term planning.

Having an accurate understanding of imbalances by

class size across colleges, departments, or majors

could help institutions pinpoint areas that need

improvement. In addition, institutions could examine

whether access to smaller classes is inequitable

across certain student groups, such as among

minority, first-generation, or first-year students.

We will argue that the current definition of class size

used in the CDS Initiative (2018) is insufficient for

both internal planning and external consumption.

As institutional researchers, it is our job to provide

meaningful and accurate information to both

internal and external parties. While the traditional

measure of class size may be accurate, this paper

describes ways in which we could provide data that

are more meaningful. Institutional researchers and

higher education professionals have an obligation

to update this metric to reflect the actual student

experience.

DEFINING CLASS SIZEBased on the conventions of the CDS,

undergraduate class size is calculated based on

the number of classes with a given class size range.

Classes are divided into sections and subsections.

A class section is an organized class that is offered

by credit, is identified by discipline and number,

meets at one or more stated times in a classroom or

setting, and is not a subsection such as a laboratory

or discussion section. A class subsection is a part of

a class that is supplementary and meets separate

from the lecture, such as laboratory, recitation, and

discussions sections. In calculations of class size,

we count only the sections of a class and discard

the subsections. The CDS conventions consider

any section or subsection with at least one degree-

seeking undergraduate student enrolled for credit

to be an undergraduate class section, but exclude

distance learning, noncredit, and specialized one-

on-one classes such as dissertation or thesis, music

instruction, one-to-one readings, independent study,

internships, and so on. If multiple classes are cross-

listed, then the set of classes are listed only once to

avoid duplication (CDS Initiative, 2018). This means

that we count stand-alone classes, defined as having

only one component, once per section in the class

section portion.

For classes with multiple components, such as a

lecture section combined with a lab or discussion

section, we count each lecture section once in the

class section portion while we count each associated

lab or discussion section once in the class

subsection table. In traditional class size metrics,

the CDS counts only the class section portion of the

class while the CDS discards the subsection from the

calculation. This metric is relatively easy to compute

and is comparable across campuses, but it may not

be representative of the student experience.

We define “student experience class size” as the

percent of time spent by a student in classes of

various sizes, using credits as a proxy for time.1

Calculations for this metric will be discussed later in

1. We assume that each credit associated with a class is approximately 50 minutes of class time. While this pattern is not universal across U-M, the calculation is easy to make and should be readily accessible in any institution’s data warehouse. A more complicated, but 100% accurate approach, would be to use the day and time location to derive the true number of minutes spent in each section. These data might not be accessible and would require substantial coding to calculate. We tested both approaches and our simplified approach did not meaningfully differ.

8Fall 2019 Volume

this paper. Figure 1 shows the difference between

the CDS method and our new student experience

method of computing class size. Sources that are

often used as references for prospective students

and institutions, such as U.S. News and institutional

websites, draw data from the CDS. The CDS metric

describes the share of classes in a given range

rather than the share of time spent in classes of

varying sizes. According to the U.S. News and the

U-M websites, 84% of classes at U-M have fewer

than 50 students, and 57% have fewer than 20

students. However, using our student experience

class size metric, only 19% of a student’s classroom

time is spent in classes with fewer than 20 students

and nearly 30% of their time is spent in classes with

at least 100 other students. Why do these metrics

differ so drastically?

There are three primary reasons driving these

differences. First, the traditional measure for class

size is not weighted by the number of students

enrolled. A 500-student section and a 5-student

section both count as one class, even though

many more students experience the larger section.

Second, classes are not weighted by the number of

credits associated with the class. A class worth five

credits counts for the same as a class worth one

credit, even though students likely spend five times

as much time in the first class. Finally, the traditional

measure does not incorporate subsections. It is

typical for large lecture classes to have multiple

components, such as a large lecture of 200 that

meets for 2 hours per week and 10 associated small

discussion groups of 20 students each that meet

for 1 hour per week. Students spend 67% of their

classroom time in a large lecture and 33% of their

time in a small discussion, but the traditional metric

Figure 1. Class Size by Various Sources

9Fall 2019 Volume

counts only the lecture portion. This means the

200-person lecture counts as one class, ignoring the

subsections. Our new student experience class size

metric accounts for these three factors, as will be

explained in detail in the methods section. This new

metric provides a more accurate representation of

the student experience within the classroom.

Given the immense difference between the

traditional class size measure and our student

experience version, it is only natural to question why

institutions have not moved to a different calculation

of class size. To be clear, it is not the authors’ belief

that institutions are purposely trying to push an

inaccurate measure of class size. The traditional

class size measure does hold value in describing the

number of classes available to students of various

class sizes. U-M students can choose from many

small classes and could theoretically construct a set

of classes to minimize the amount of time spent in

large classes. In reality, though, several forces could

make it difficult for institutions to switch to a student

experience version of class size.

First, there are serious consequences in rankings

and optics for many institutions, particularly larger

ones. U.S. News currently provides points to

institutions based on the share of small sections (19

and fewer students), partial points for the share of

medium sections (20–49 students), and no points for

large sections (50 or more students).2 Universities

that use larger class sizes to teach a large number

of students will see their rankings negatively

impacted if classes were weighted by the number

of students taking the class. The shift in class size

would also create poor optics for prospective

students and may impact whether they choose one

institution over another. The same would be true

for a student experience measurement of instructor

type, where larger research institutions are much

more likely to use graduate students as instructors.

In addition, having non-tenure track instructors

primarily responsible for teaching large classes, and

therefore many students, could have bad optics

for institutions. Therefore, there is a disincentive

for a single college, or a small group of colleges, to

recalculate their class size based on the student

experience. The exception would be institutions

that uniformly have very small classes, such as

small liberal arts colleges, that would see little or no

change in their calculation of class size.

A second difficulty is measuring class size from the

student perspective. Leaders of the CDS Initiative

already consider the measurement of class size to

be the second-most difficult part of the CDS, with

only calculations of financial aid deemed more

difficult (Bernstein, Sauermelch, Morse, & Lebo,

2015). At U-M, measures required to recalculate

class size to the student perspective are readily

available and clean, and require little manipulation

to combine. Institutions vary significantly in their

data capacity and availability of staff to adjust the

CDS measures. Given these challenges, the authors

of this study still believe that shifting to a student

experience version of class size would provide many

benefits, including an accurate representation of

the amount of time students spend in classes of

varying sizes and with various instructor types. This

shift would be beneficial for institutions for planning

purposes as well as for prospective students as

they weigh various institutions during the selection

process.

2. This recently changed from a system that provided points for small courses and penalties for large courses in an effort to minimize gaming of the system (Supiano, 2018). However, there is no evidence provided to back up this claim. Regardless of whether the new or old system is used, having a larger number of small classes is rewarded, and the metric is at the course level, which does not appropriately describe the student experience.

10Fall 2019 Volume

METHODSThe purpose of this study is to create a student

experience version of class size. We drew data

from the U-M data warehouse, specifically from the

Learning Analytics Data Architecture (LARC) and

College Resource Analysis System (CRAS).3 LARC is a

flattened, research-friendly version of the raw data

warehouse that houses data about students, their

background, their progress, and their coursework.

CRAS is a data warehouse system that houses data

about classes and the instructors that teach them.

The sample included first-time freshman students in

cohorts between 2001 and 2012, examining classes

taken within 4 years of entry. Freshman cohorts are

between approximately 5,500 and 6,500 students

during this period. Individual study classes and one-

on-one classes were removed from the sample, as

were classes with no CRAS information, including

subjects such as medicine, dentistry, armed forces,

study abroad, and classes through the Committee

on Institutional Cooperation program. The final

sample included 70,426 first-time students and

3,398,320 class sections taken in these students’ first

4 years of study, between 2001 and 2016.

Calculating Class Size

As previously noted, we made three adjustments to

the traditional measure of class size: (1) weighting

for number of students, (2) weighting for credits

associated with the section, and (3) incorporating

subsections. Table 1 provides an example of how

3. LARC is unique to U-M, although some institutions have developed a similar research-friendly database. CRAS is also unique to U-M because there are some calculations made with institution-specific formulas. However, these databases comprise data that are regularly available (although not always clean) at all institutions, such as student and class information, classes taken, and the number of students in a given class.

Table 1. Example of Distribution of Class Credits in Our Framework

Number of Students Class 1 Class 2 Class 3 Class 4

Total Credits

% of Total

2–9

10–19

20–29 1 (Lab) 1 (Discussion) 2 17%

30–39

40–49 3 (Lecture) 3 25%

50–99 2 (Lecture) 2 17%

100–199 2 (Lecture) 2 17%

200+ 3 (Lecture) 3 25%

Total Credits

3 4 3 2 12 100%

11Fall 2019 Volume

we accounted for credits and subsections in our

student experience framework.

This example shows a hypothetical student’s

coursework for one term. This student took four

classes in this term for a total of 12 credits. Classes

1 and 2 had multiple components, with a lecture

and either a discussion or a lab. We divided credits

for these classes among the section and subsection

based on the amount of time or credits associated

with each component. Then we put each component

into the appropriate class size bin, shown in the left

column. Classes 3 and 4 were stand-alone classes

that contained only a lecture component, so there

was no need to divide their credits. Once we had

distributed all the credits for each class to the

appropriate class size bin, we totaled credits across

each class size row. In doing so, we accounted for

credits and the multicomponent nature of classes,

fixing two of the issues with the standard definition

of class size. The third piece relates to weighting

class size by the number of students in the class. In

this framework, we theoretically create a table like

this for each student and each term the student

attends. We then summed across every student

and term. Since the level of observation is a class

enrollment, we naturally weight by the number

of students in the class because there will be 50

observations if there are 50 students in a class, or

5 observations for a class with 5 students. A basic

assumption made by this framework is that one

credit is equal to approximately 1 hour of class

time. While there are a small number of classes that

violate this assumption, we do not believe it would

impact our overall results in a meaningful way. It is

also important to note that enrollments for cross-

listed classes were combined into the home class.

At U-M, if there are multiple cross-listed sections,

one is considered the home class and the rest are

considered away classes. This means that if there

are three cross-listed sections with 12, 15, and 18

students, our data would show one class with 45

students.

Once we calculated the percent of time (using

credits as a proxy) that a student earned in various

class sizes across his first four years, we calculated

percentiles for each enrollment group across the

entire university by college, by major, and by year

in school.4 College and major are determined by

the last college or major associated with a student.

If a student graduated with a bachelor’s degree, we

used his graduating college or major. If a student

departed prior to completing his degree, we used

the last known college and major associated with

him.5 Rather than showing just the median or mean,

we chose to use five percentiles (10th, 25th, 50th,

75th, and 90th) to show the distribution of student

experiences. Finally, we mapped these percentiles

into figures that show the range of student

experiences for a given class size.

CASE STUDYThis section will examine how class size varies

across U-M. The figures in this section represent

the distribution of time that students spend in

classrooms of varying sizes. Class size was grouped

into eight bins of varying sizes to create a smooth

figure and to mimic the traditional measure of

class size. Figure 2 shows the distribution of class

size across the entire university. The black line

represents the median student, the dark gray

shaded area represents the 25th to 75th percentile,

and the light gray shaded area represents the 10th

4. Results show the 10th, 25th, 50th, 75th, and 90th percentile for each enrollment group. The term “college” refers to an academic college, such as engineering, liberal arts and science, or education.

5. We considered removing these students but decided that doing so could introduce some selection bias. For example, students that drop out may select larger courses; if we remove them, we may distort the student experience.

12Fall 2019 Volume

to 90th percentiles.

To interpret this figure, consider the enrollment

group of 200+ students. The black line indicates that

the median U-M student spent about 15% of her

time in classes with 200 or more students. Moving

above the black line to the edge of the dark gray

area, we see that about a quarter of students spent

about 23%–24% of their time in classes with 200 or

more students. The outer edge of the light gray area

indicates that 10% of students at U-M spent more

than 30% of their time in these very large classes. If

we move to the 20–29 enrollment group, we see that

the median student spent about 20% of her time

in classes with 20–29 students. There is a very wide

range of experiences (25%) between the 10th and

90th percentiles, indicating that students may spend

vastly different amounts of time in classes with

20–29 students. The spread is only about 10% wide

for enrollment groups between 30 and 49 students,

indicating there is less variability in the percent of

time spent in medium-size classes. Overall, it is clear

that students’ time is more heavily weighted in both

large (50+ students) and small (10–29 students)

classes. This means that students spend their time

in classes of varying sizes, but classes at U-M tend to

favor high or low enrollments on average. This also

implies that very large classes likely have a smaller

enrollment component tied to them, such as a

discussion or lab section.

At a university the size of U-M, with an

undergraduate population of almost 30,000, it

is natural to assume that student experiences

may vary greatly across the institution, such as by

college, major, or year in school. Figure 3 shows

the distribution of class size in the College of

Engineering, which we chose because it is the

Figure 2. Percent of Time Spent by Size Across the University of Michigan

13Fall 2019 Volume

second-largest college on campus and differs

significantly from the trends for the median student.

Once again, the black line and gray shaded areas

represent the percentiles for students in a given

college. We added the gray dotted line here to show

the median U-M student from Figure 2, allowing

us to examine how the college differs from the

overall pattern at the university. In the College of

Engineering students tend to take coursework with

very different class sizes compared to the median

U-M student. In particular, engineering students

take fewer small classes (10–29 students) and

very large classes (200+ students), and instead

take more classes in the middle range (30–199

students). Part of the difference could be attributed

to deliberate planning by the College of Engineering

and part could be related to the size of classrooms

in the engineering buildings. Classroom caps,

and subsequently enrollment caps, for classes

in engineering tend to lie in the middle of the

enrollment group distribution.

While not shown, we created figures for every

college on campus. The trends of these figures show

that class size differs significantly across colleges.

The College of Literature, Science, and the Arts

(LSA) has very similar trends to the median U-M

student, in part because it is the largest college on

campus. Given that more than 60% of students

are in LSA, this college drives much of the median

class size. As one would expect, smaller and more

narrowly focused colleges such as the College of

Architecture and Urban Planning, the College of

Art and Design, and the College of Music, Theatre,

and Dance had significantly higher levels of small

classes (2–29 students) and fewer very large classes

(200+ students). The College of Public Policy tends

to mimic the trends of LSA, in part because students

Figure 3. Percent of Time Spent by Enrollment Group in the College of Engineering

14Fall 2019 Volume

spend their first two years in LSA before declaring

their major. The College of Business had very high

levels of medium-size classes (40–99 students)

because many of its core classes have enrollments

of 40 to 80 students.

Class size also varies in systematic ways by

major, even within a college. Comparing the

major of Arts and Ideas in the Humanities, a

small, multidisciplinary major, to the major of

Biopsychology, Cognition, and Neuroscience (BCN),

a large, premed major, yielded large differences in

class size. The median Arts and Ideas major spent

about 50% of her time in classes with 20 or fewer

students, which is twice the 25% of the median LSA

student. Students in the BCN major, on the other

hand, took a much larger share of large lectures,

rising out of the 90th percentile for LSA students.

They spent nearly 33% of their time in classes with

more than 200 students, compared to only 20% of

the median LSA student’s time. While we observed

systematic differences between majors, we also

found that there were some majors that had very

similar class size structures.

A final way to observe how class size varies across

an institution is by comparing class size by academic

level. Rather than aggregating across a college

or major, we aggregated by a student’s year in

school (e.g., freshman, sophomore, junior, senior).

Students in their first year typically fulfill their

general education requirements, which tend to be

classes taught to many students at once. By their

junior or senior year, students tend to take many

classes within their major of increasing depth and

specialization, characterized by smaller class sizes.

Splitting the data in this manner did yield interesting

differences over time. As shown by the dotted line in

Figure 4. Class Size Within the College of Engineering by Academic Year

15Fall 2019 Volume

Figure 4, students in the College of Engineering took

similar coursework to LSA students in their first year

compared to the median U-M student from Figure

2, taking mainly classes consisting of large lecture

and small discussion groups. This is likely because

students are fulfilling their general education

requirements. By their third year, the solid black line

indicates the median engineering student deviates

from the pattern at the college level and takes a

much higher proportion of classes with 50–100

other students; these classes comprise classes in

the students’ major.

Overall, we created figures for every major, college,

academic level, and student across campus to

examine how class size varied across U-M. The

median student within some majors had class size

distributions that were very similar to the median

student in their college and the university, but many

majors differed significantly from the general trends.

Similarly, there were some students within majors

that varied significantly from the median student in

their major. These figures show that class size can

vary significantly across an institution. A department,

college, or institution can use this information to gain

a more nuanced understanding of the experiences

of their students. For example, if students that

tend to take only large lectures or small discussion

classes perform worse, then an institution could

adjust its advising to promote students to take

classes of varying sizes. Similarly, an institution could

identify whether certain student groups, such as

first-generation students, may benefit from a more

intimate classroom environment where they receive

more attention from instructors.

HOW SHOULD WE MEASURE CLASS SIZE?This paper has shown that the traditional measure

of class size is not sufficient if it is meant to provide

information about a student’s actual experience

in the classroom. Previous research has shown

that increasing class size has a mixed but generally

negative impact on student learning and satisfaction.

Accurately measuring class size is also important

to institutions because it impacts productivity. For

example, increasing or decreasing the class size of

introductory calculus, a class that most students on

campus take, can have vast implications for the cost

of the class. If an institution wanted to recalculate

class size with the student experience at the core,

what would it look like?

We will first consider two simple adjustments:

weighting for credits and weighting for students.

Table 2 shows the distribution of class size given

four different calculations and Figure 5 provides

a visual representation of the table. The first

calculation is the traditional measure of class size,

with no adjustments. This means there is one

observation per lecture. The second column weights

the traditional measure by number of credits. A

class of four credits is now worth twice as much as

a class of two credits. This slightly shifts the class

size distribution down because large class sections

have corresponding subsections. Consider a class

worth three credits, two earned in a lecture and one

earned in a lab. Since the lab, or subsection, is not

counted in the traditional measure, one of the three

credits is discarded, deflating the value of a large

class. The third column accounts for subsections

and credits, appropriately distributing all the credits

associated with each class. A class worth three

credits including a lecture and a lab, as described

above, is now fully included in the metric for class

16Fall 2019 Volume

size (Column 3). This lowers class size considerably

because we are not removing the subsection but

instead including it in our calculation. Subsections

tend to be smaller, which drives the distribution

down.

The fourth column shows class size distribution if

we weighted only for the number of students in a

class. Using this calculation, a large lecture of 500

students is counted 500 times, while a small class

of 5 students is counted only 5 times. As expected,

this dramatically shifts the distribution of class size

upward, nearly flipping the distribution of class size

from the traditional measure. Column 5 accounts

for weighting by credits and students, and includes

subsections simultaneously. This gets closer to

the perfect measure of the student experience

described earlier in this paper and vastly improves

the understanding of class size for institutions

and potential students. Figure 5 shows a visual

representation of each strategy for calculating

class size. We can see that the traditional measure

(nonstudent experience), weighting for credits, and

weighting for subsections are very similar, with

slight increasing low enrollments (fewer than 30)

and decreasing other enrollments (30–200+) after

accounting for subsections. Once we weight for

students there is an immediate shift, nearly flipping

the distribution. However, this shift goes too far;

accounting for credits and subsections provides a

balance between the three strategies.

It is important to note that our calculation of class

size used student-level micro data to account for

these changes, in part due to other related research.

However, this calculation may be burdensome for

institutions. A simpler way to achieve the same

results would be to take the class-level data used for

Table 2. Class Size by Three Different Calculations

Class Size

Traditional Measures

(1)

Weight for Credits

(2)

Account for Subsections

(3)

Weight for Students

(4)

Weight for All Changes

(5)

2–9 13.9% 14.1% 11.4% 2.0% 2.5%

10–19 32.1% 32.5% 31.1% 13.0% 15.4%

20–29 23.0% 23.6% 34.0% 14.2% 21.9%

30–39 8.2% 8.5% 9.4% 6.8% 9.3%

40–49 4.4% 4.3% 3.2% 4.6% 4.4%

50–99 10.5% 10.2% 6.4% 18.2% 15.8%

100–199 4.9% 4.5% 2.9% 17.2% 13.8%

200+ 2.9% 2.4% 1.7% 24.0% 16.8%

17Fall 2019 Volume

the CDS and simply weight each class section and

subsection by the number of students and credits.

This simplifies the process and limits the resources

required to pull and process the data.

LIMITATIONS OF NEW CLASS SIZE MEASUREMENTWhile this new class size metric may provide a

clearer picture of the student experience at an

institution, it does not come without limitations

and challenges. The first challenge is the

technical barriers of calculating the new class size

measurement. For example, institutions may house

the required variables in different data systems, but

replacing or adding a new metric could be difficult to

implement and could increase the reporting burden

on institutions. Most of what is required to adjust

to this new metric is already required for the CDS

version of class size. Institutions must calculate the

number of sections and subsections of a given class

size. This means they must know the exact class size

for every section and subsection. The only missing

piece is to allocate time between the section and

the subsection. At U-M, a field for distributed hours

is contained in the data warehouse, but that may

not be the case for all institutions, some of which

may not even use the Carnegie credit system. For

these institutions, a calculation could be made to

allocate credits based on the amount of classroom

time for each section and subsection. For example, a

three-credit class that meets for 2 hours in a lecture

section and 1 hour in a discussion section could split

the class into two credits for the lecture and one

Figure 5. Class Meeting Size Distribution by Five Different Methods

18Fall 2019 Volume

credit for the discussion. This would require some

up-front work to create these calculations based on

day/time information, but is not unmanageable.

The second set of challenges relate to the nuance

that is inherent in the new metric. Presenting

students with a single metric for an entire university

is simple and easy to explain but leaves out a lot of

nuance. If an institution were to recreate the figures

in this paper, it would introduce some challenges

for interpretation. As shown in Figure 3, there are

stark differences in class size between engineering

and LSA students; the same differences can be

shown between majors or year in school. The large

number combinations and comparisons by college,

major, and year in school present challenges when

trying to show or explain the data to prospective

or current students. While a single, institution-wide

metric leaves out nuance, it is a vast improvement

over the current metric used by institutions and

rankings. If an institution wanted to provide more

nuance, it could create an interactive dashboard

for prospective and current students to view the

nuanced version of class size presented in the

figures of this paper. Students could select a college

or major from a list to see what the distribution

of class size looks like for students in that major.

Institutions could pare down these figures to

show only the median for simplicity or to provide

a detailed explanation of how the percentiles

work when students first use the new dashboard.

However, with a nuanced view an institution could

also show students that class size may be lower for a

given major, or how class size may lower as students

progress toward their degrees.

DISCUSSION AND FUTURE WORKAs institutional researchers it is imperative for us to

provide data that are both accurate and meaningful.

This paper argues that the traditional measure of

class size is not a meaningful representation of

what students experience in the classroom. The

traditional measure of class size illustrates only the

proportion of classes that are small, not the amount

of time that students spend in small classes. This

is problematic for prospective students, who could

use meaningful class size data to determine where

they want to attend college. It is also problematic

for institutions, which may not understand the

extent of large or small classroom experiences on

their campuses. While limited, previous research in

higher education suggests that class size matters for

student outcomes and satisfaction in classes.

This paper suggests that the measurement of class

size could be altered by weighting for the number

of students and credits associated with the section,

and accounting for subsections. We suggest that

institutional researchers consider revamping their

class size metric to reflect the student experience

in the classroom more accurately. Nearly all the

required components (number of students in

each class and section/subsection, and number of

classes) for this calculation are used by the current

CDS metric. We believe that distributed credits, the

potential missing component, is likely captured and

readily available at many institutions, which would

make this adjustment relatively easy. While this could

require an investment of time on behalf of some

institutions, we believe the potential benefits will

outweigh the investment. At a minimum, we suggest

that institutional research professionals consider

reweighting their current metric of class size by the

number of students in each lecture section. This new

19Fall 2019 Volume

metric provides a more accurate description of class

size from the student experience.

Future work on this topic could yield more

improvements in the description of the student

experience at institutions of higher education. First,

the range of class size also varies significantly across

colleges and departments, indicating that a simple

institution-wide metric masks important differences

for students who plan to major in different fields.

Institutions could take this idea one step further

to create an interactive dashboard that allows

prospective and current students the opportunity

to see the range of class size experiences for majors

in which they have interest. A second improvement

institutions could make would be to pair data

about class size and instructor type (e.g., IPEDS

instructor type). By combining the percent of time

spent in varying class sizes and varying instructor

types, students will gain a clearer picture of what

their classroom experience would be at a particular

institution.

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Bernstein, S., Sauermelch, S., Morse, R., & Lebo,

C. (2015). Fundamentals and best practices for

reporting Common Data Set (CDS) data. Presented

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Taylor, E. (2017). The effects of class size in online

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Bettinger, E. P., & Long, B. T. (2018). Mass instruction

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Chetty, R., Friedman, J. N., Hilger, N., Saez, E.,

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Dee, T. S., & West, M. R. (2011). The non-cognitive

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SmallClassExecSum.pdf

21Fall 2019 Volume

Rex Gandy, Lynne Crosby, Andrew Luna, Daniel Kasper, and Sherry Kendrick

About the Authors

The authors are with Austin Peay State University.

Abstract

While Markov chains are widely used in business and industry, they are used within higher education

only sporadically. Furthermore, when used to predict enrollment progression, most of these models use

student level as the classification variable. This study uses grouped earned student credit hours to track

the movement of students from one academic term to the other to better identify where students enter or

leave the institution. Results from this study indicate a high level of predictability from one year to the next.

In addition, the use of the credit hour flow matrix can aid administrators in identifying trends and anomalies

within the institution’s enrollment management process.

Keywords: Markov chains, enrollment projections, enrollment management, enrollment trends, enrollment

Enrollment Projection Using Markov Chains: Detecting Leaky Pipes and the Bulge in the Boa

The AIR Professional File, Fall 2019

Article 147

https://doi.org/10.34315/apf1472019

© Copyright 2019, Association for Institutional Research

22Fall 2019 Volume

INTRODUCTIONThe current challenges facing higher education

administrators create myriad reasons to find

a crystal ball of sorts to effectively forecast

enrollments, predict how many current students will

stay at the institution, forecast new students, and

adequately estimate revenues. These challenges

have become only more pressing in recent years.

More than 20 years ago, when public college and

university revenues were ample, administrators

were not readily concerned about the future of

college enrollments or student persistence. State

appropriations were healthy and usually made up

more than half of an institution’s revenue source.

Moreover, with lower tuition more students could

afford to obtain a degree without going into

significant financial debt (Coomes, 2000).

The costs to run higher education have skyrocketed,

however, causing today’s institutions to seek scarce

resources within an ever-diminishing financial

pool. As states tackle other pressing issues such

as infrastructure, entitlements, and prisons, the

amount they give to higher education naturally

wanes. Decreased state revenue, therefore,

compels institutions to increase tuition to make

up the difference. According to Seltzer (2017),

for every $1,000 cut from per student state and

local appropriations, the average student can be

expected to pay $257 more per year in tuition and

fees. He further notes that this rate is rising.

In addition to decreases in state revenues, higher

education administrators are under increasing

pressure to be accountable to federal and state

governments as well as to regional and discipline-

based accreditors. This accountability is increasingly

seen in tougher reporting standards, outcomes-

based funding formulae, and mandated student

achievement thresholds.

The closest resource to a crystal ball available to

administrators is a set of mathematical prediction

tools. These prediction tools range from simple

formulae contained in spreadsheets to much more

complicated regression, autoregressive integrated

moving average (ARIMA), and econometric time

series models.

According to Day (1997), current predictive tools

that are statistically based rely on the institution’s

ability to access and manipulate large datasets

and individual student-record data. While more-

complicated statistical models incorporate variables

such as tuition cost, high school graduate numbers,

economic factors, and labor-market demand,

other models look more specifically at institutional

indicators such as high school grade point averages

of entering freshmen, as well as the retention,

progression, and graduation rates of students.

One such model, the Markov chain, has been

relatively underutilized as an enrollment projection

tool in higher education. When used properly,

however, it can aid institutions in determining

progression of students. Specifically, Markov chains

are unique from more-traditional ARIMA and

regression prediction tools in that the following is

true:

1| Markov chains can give accurate enrollment

predictions with only the previous year’s data.

These predictions can be helpful when large

longitudinal databases are not available.

2| They can generate predictions on segments of

a group of students rather than on the entire

population. Other models often require the use

of the entire population.

23Fall 2019 Volume

3| The almost intuitive nature of the Markov

chain lends well to changes in student flow

characteristics that often cannot be explained

by a complex statistical formula.

Moreover, Markov chains might be particularly

helpful in determining progression of students

during benchmark years when enrollments vary

significantly due to state mandates and policies,

or due to institutional changes in admission

standards. The purpose of this study is to show how

a Southeastern, masters-level (Larger Programs)

public institution utilized the unique properties of

this model to create a tool to better understand

credit hour flow and student persistence.

Enrollment Management’s Problem with Leaky Pipes and the Bulge in the Boa

While enrollment management has clearly evolved

since the inception of the field of enrollment

management in the 1970s, some fundamental

processes have essentially stayed the same.

Institutions have always wanted to attract the right

students who fit well within the institution’s role,

scope, and mission. Once matriculated into the

institution, there is also a strong desire for students

to adequately progress through their program

and graduate within a reasonable amount of

time (Hossler, 1984). As enrollment management

developed through time, however, administrators

became increasingly aware that college-age students

were more difficult to enroll, higher tuition was

causing some students to forgo their degree,

and institutional loyalty was waning as students

transferred to similar or different institutions.

Furthermore, institutions have seen an increasing

number of students who are not fully prepared for

the rigors of college work, putting greater enrollment

strain on institutions (Johnson, 2000).

After more than 40 years of enrollment management

within higher education, it is not surprising that

metaphorical associations have entered the lexicon

of the profession as administrators try to better

understand and predict student matriculation,

persistence, and graduation. For instance, Ewell

(1985), referred to students progressing and

moving throughout their program as student flow,

while Clagett (1991) discussed following the flow

of student cohorts through to graduation. Luna

(1999) used the concept of student flow to explain

the various pathways by which the institution may

retain students, and Torraco and Hamilton (2013)

discussed the student flow of selected groups of

minority students. Furthermore, many software

companies have exploited the student flow

metaphor to describe use of data to identify areas

where leakage is present in student flow pipelines. It

is easy, then, to see how the management of student

retention can be associated with a pipeline and how

administrators are busy trying to plug the leaks.

Markov chains are uniquely suited to identifying

these leaks because they can model student flow

as a set of transitions between several states, much

like a set of pipes with various inflows, outflows,

and interconnections. In addition to using the

model to project enrollments, it is also possible to

observe from year to year where students enter the

absorbed state (i.e., do not return to the institution).

Leakage within the student credit hour (SCH) flow

pipeline occurs when students withdraw or stop out

due to reasons that are academic, nonacademic, or

both. If the model can isolate where the major leaks

occur, the institution can identify causes and work

to retain and maintain the flow of students within

the pipeline. These leaks in the student flow pipeline

can be detected and monitored from term to term

so that the institution can develop strategies to

maintain a healthier flow.

24Fall 2019 Volume

Another colorful bit of jargon among enrollment

management professionals is the idea of

bulging enrollments. For example, Fallows and

Ganeshananthan (2004) use the term “bulging

of enrollments” to describe a significantly larger

share of students needing financial aid or when,

due to rising tuition costs, students bulge into

less-expensive 2-year colleges. Herron (1988) uses

the term “bulge in the boa” to define instances of

oversupply in student populations quickly entering

the student flow pipeline, much as a boa constrictor

swallows a large meal. Liljegren and Saks (2017)

added that these bulges can significantly affect

higher education and its future. These bulges occur

when large groups of students suddenly enter

higher education, putting a strain on the student

flow pipeline. As the bulge dissipates, its effects

may remain, and it may redefine student flow for

the future. With Markov chain models, institutions

can monitor these bulges in the system so that they

can address issues such as course offerings and

instructor availability.

Markov Chains and Higher Education

A Markov chain is a type of projection model

created by Russian mathematician Andrey Markov

around 1906. It uses a stochastic (random) process

to describe a sequence of events in which the

probability of each event depends only on the state

attained in the previous event.

The Markov chain is a stochastic rather than a

deterministic model. Unlike a deterministic process

where the output of the model is fully determined by

the parameter values and by sets of previous states

of these values, a stochastic process possesses

inherent randomness: the same set of parameter

values and initial conditions can lead to different

outputs.

Take, for example, the scenario of an individual

returning home from work. In a deterministic

process, there is only one route (Route A) from

work to home, and the amount of time to get home

depends only on the variable speed of the driver. In

a stochastic process, the individual will have multiple

routes (Routes A, B, and C) from which to choose,

and each of the routes intersects the other routes

at various points. The randomness of the process

occurs when the individual combines routes to go

home, if she makes the choices at each intersection

randomly. For example, the driver may take Route A

part of the time, followed by Route C, then Route B,

and back to Route A again, or take some completely

different path. There are many random possibilities

the individual may take to get home, leading to a

variety of possible driving times.

Markov chains utilize transition matrices that

represent the probabilities of transitioning from

each possible state to each other possible state.

These states can be absorbing or nonabsorbing:

nonabsorbing states allow future transitions to other

states while absorbing states do not.

Markov chains have been widely and successfully

used in business applications, from predicting sales

and stock prices to personnel planning and running

machines. Markov chains also have been used in

higher education, albeit with much less frequency.

In most studies where Markov chains were used in

enrollment management, the various transitional

states were categorized either by student

classification or by other simpler dichotomous

measures. Given the strength of the Markovian

stochastic process in generating student flow

probabilities using data only from the previous

year, the process of classifying students into other

kinds of states could be appealing. Such states

25Fall 2019 Volume

could include SCHs, student debt, and (on a more

systemwide level) the transitioning from one

institution or program to another. The possibilities

are diverse.

One of the first to use Markov chains in determining

enrollment projections was Oliver (1968) when

he compared Markov chains to the much more

established use (at that time) of grade progression

ratios to predict enrollments at the University of

California. According to Oliver’s study, enrollment

forecasting made a prediction on the basis of

historical information on past enrollment and

admission trends. In determining a stochastic

process, Oliver demonstrated that the fraction of

students who leave one grade level (class status) i

and progress to class status j is a fraction pij; that

progress could also be time dependent. These

fractions pij can also be interpreted as random

transition probabilities. He determined that the

process allowed for contributions in one grade level

that were identified by their origins, such as prior

grade level, returning to the same grade level, and

new admissions (Oliver, 1968).

According to Hopkins and Massy (1981), the use of

Markov chains allows the researcher to observe the

flow of students from one classification level (i.e.,

freshman, sophomore, junior, senior) to the next

class level. The chain also incorporates students who

stay at the same class level from one year to the

next. Therefore, the Markov chain for class level, as

studied by Hopkins and Massy, can be described as

follows:

1| The number of students in class level i who

progress to class level j

2| The number of students in class level i who stay

in the same level

3| The number of students who leave the

institution (drop out, stop out, or graduate)

Similarly, Borden and Dalphin (1998) used Markov

chains to develop a 1-year enrollment transition

matrix to track how students of each class level

progressed. The authors found that unique Markov

chain models were valuable in measuring student

progression without having to rely on 6-year

graduation rate models, which could be ineffective

due to the large time lags. Specifically, the model

was built around a transition matrix where student

flow was tracked from one year to the next, and the

rates of transition from four nonabsorption states

(i.e., freshman to sophomore) were placed into a

matrix that was separate from the two absorption

states (i.e., drop out, graduate).

Using the percentages in the two matrices, those

students who continue in nonabsorption states were

processed through the matrix using the established

rates of transition until, asymptotically, all students

reach the final absorption state.

Additionally, Borden and Dalphin (1998) developed

discrete Markov chain processes to simulate

the effect of changes in student body profile on

graduation rates. In these models, the authors

incorporated credit-load and grade performance

categories. Their results indicated that, while

there was a strong association between grade

performance and persistence, it took very large

changes in levels of student performance to impact

retention and graduation rates modestly.

In a more narrowly focused study, Gagne (2015)

used Markov chains to predict how English

Language Institute (ELI) students progressed

through science, technology, engineering, and

math (STEM) programs. Specifically, the model

26Fall 2019 Volume

created transitional (nonabsorbing) states based

on classification level and three absorbing states to

include those students who left the institution, those

who graduated from a STEM program, or those who

graduated from a non-STEM program. Findings from

their study indicated that the ELI students tended to

progress at a higher rate than non-ELI students in

STEM programs, and that ELI students who repeated

the freshman year were more likely to repeat again

than they were to transition to the sophomore year.

Correspondingly, a study by Pierre and Silver (2016)

used Markov chain models to determine the length

of time it took students to graduate from their

institutions. As with previous studies, students

were divided into nonabsorbing transitional states

(i.e., freshman, sophomore, junior, senior) and

absorbing states (i.e., graduate, nonreturning). Using

the Markovian property, the future probability of

transitioning from one state to another depended

only on the present state of the process and was not

influenced by its history. The study found that it took

5.9 years for a freshman to graduate and 4.5 years

for a sophomore to graduate from the institution.

Brezavšček, Bach, and Baggia (2017) successfully

used Markov chain models to investigate the pattern

of students’ enrollment and academic performance

at a Slovenian institution of higher education. The

model contained five transient or nonabsorbing

states and two absorbing states. The authors used

student records for a total of eight consecutive

academic seasons, and estimated the students’

progression toward the next stage of the program.

From those transition percentages they were able

to obtain progression, graduation, and withdrawal

probabilities.

As mentioned earlier, most Markov chain models

involving enrollment management and prediction

use student classification to create the various

states of the model. Using student classification in

model specification, however, could create states

that are overly broad in nature since, at most

semester-based colleges and universities, student

classification varies by 30 hours.

Ewell (1985), who also used Markov chains to

predict college enrollments, noted two limitations

of the models. First, because the estimation of the

probabilities rests on historical data, Markov chains

may be sensitive to when the data were collected.

This could be especially true with significant

enrollment gains or declines from one year to

the next. Second, according to Ewell, different

subpopulations may behave in different ways, thus

necessitating the need to disaggregate into smaller

groupings.

However, the Markov chain’s attributes may allow

a unique ability to detect the leaks and bulges.

Because this type of projection model uses the

stochastic process to describe a sequence of events

in which the probability of each event depends only

on the state attained in the previous event, changes

to student flow are immediate and are not subject to

potentially skewed results of the past. In short, the

limitations mentioned by Ewell (1985) can be utilized

when building the student flow matrices to detect

significant shifts in enrollment and to determine

which groups of students are leaving the institution

at a higher rate.

METHODOLOGYThe current study used Markov chains to predict

Fall enrollment at a Southeastern, masters-level

(Larger Programs) public institution based on

annual Fall semester enrollment for degree-seeking

27Fall 2019 Volume

undergraduates. The process involved obtaining

data from the institution’s student information

system and separating students into groupings

based on their cumulative SCHs earned. Student

flow was measured from Fall of year i to Fall of year

i+1 based on whether students stayed within their

credit hour category, moved into another credit

hour category, or did not enroll at the institution.

These student flow changes for each category were

then summed and applied to year i+2 as a prediction

of enrollment.

Within the model, at a given point in time each

student has a particular state, and each student

is treated as having a particular probability of

transitioning to each other state or staying within

the same state. Most of these states are based on

the number of SCH the student has accumulated

(i.e., the SCH category). Because the SCH category

of a student was determined by the number of

cumulative SCHs a student earned, most of the

credit hour flow scenarios included students

advancing to a higher credit hour category or

students withdrawing or graduating. While it is rare

for a student to move from a particular credit hour

category to a lower category, it can happen through

the transfer process when, after the student has

enrolled, the current institution does not accept

certain SCHs from the former institution.

The characteristic that makes this model a Markov

chain is the fact that a given student’s transition

probabilities between states are assumed to depend

only on that student’s current state and not on any

of the student’s previous states. This is a simplifying

assumption that allows all students within a given

state to be treated similarly regardless of their

histories. Otherwise, the model would become much

more complicated and difficult to apply.

The main parameters of the model are estimates

of these transition probabilities. These transition

probabilities are estimated by calculating the

fractions of students that transitioned from each

state to each other state relative to the number

of students initially in that state in past years’

enrollment data. The other parameters of the model

are the fractions of new incoming students by credit

hour category. The total number of new incoming

students is assumed to be fixed, thus the estimated

number of incoming students by credit hour

category follows from these fractions.

The model process is recursive in that predictions

for Fall X are produced from the enrollment data

from Fall X-2 and Fall X-1 and the subsequent flow

rates from Fall X-2 to Fall X-1.

We can now describe the basic assumptions that we

used to construct the predictive models:

1| Each model models flow from one year to the

next and is named accordingly. For example, Fall

2013 to Fall 2014 is known as the 13_14 Model

and is based on the starting data for Fall 2013

and the new student data from Fall 2014.

2| As the model is applied, the output headcount by

SCH level for the (i+1)th year becomes the input

headcount for the next iteration of the model.

3| When the model is applied to a future year, the

total number of new students is assumed to be

constant and the same as the number of new

students for the (i+1)th year. The distribution of

new students by SCH level is also assumed to be

constant.

4| When the model is applied to a future year, it

is assumed that the fractional student loss and

fractional student continuation ratios are fixed

by SCH level.

28Fall 2019 Volume

5| When the model is applied to a future year it is

assumed that the fractional flow from SCH level

to SCH level is the same as for the year used to

construct the model.

The model divides the undergraduates into 24

6-SCH groupings. This method uses historic

ratios of SCH student subsets gathered from the

student information system to predict future Fall

headcounts.

The 6-SCH groupings used in this model are

individually less broad than the more familiar

student classification levels. However, it is possible

to aggregate the 6-SCH bins into a version of these

student levels, which we define as

• Freshmen: ≤30 SCH

• Sophomore: >30 SCH and ≤60 SCH

• Junior: >60 SCH and ≤90 SCH

• Senior: >90 SCH

Note that these classification-level definitions do

not exactly match the institution’s definitions. In

using SCH groupings, the enrollment pipeline may

be much more finely observed and enrollment

patterns among students may be more precisely

distinguished. While it is the goal of this study

to develop a model to predict the coming Fall

enrollment once the previous Fall enrollment is

known, the model will not address enrollment by

major, academic department, or college.

MODEL DESCRIPTIONThe student information system parsed out students

into the various SCH categories based on the

predetermined groupings. These students were

then tracked during the following Fall semester to

determine student flow percentages. Within this

study, student flow states are defined as:

1| students in credit hour group j who stayed

within that group,

2| students in credit hour group j who moved to a

different credit hour group,

3| students in other credit hour groups who

moved to group j, and

4| students who were no longer enrolled at the

institution.

Within this model, the following terms and symbols

are used:

1| n is the number of SCH levels in the model (n =

24 for the 6-SCH groupings).

2| hij is the ith Fall semester headcount for the jth

SCH level.

3| Hi is the total undergraduate headcount for the

ith semester.

4| lij is the number of the hij subset students not

enrolled the next Fall semester.

5| Li is the total number of undergraduates

enrolled in the ith Fall semester that are not

enrolled in the (i+1)th Fall semester.

6| cij = hij – lij is number of continuing students in

the jth SCH level.

7| Ci is the total number of undergraduates that

enrolled in the ith Fall semester that are also

enrolled in the (i+1)th Fall semester.

29Fall 2019 Volume

8| dijk is the number of the continuing cij subset

students that move from SCH level j to SCH level

k from the ith Fall to the (i+1)th Fall.

9| wij is the number of the Ci subset students that

flow from all other levels into level j.

10| oij is the number of the cij subset students that

flow out of level j into all other levels.

11| s(i+1)j is the number of the new incoming students

for the (i+1)th Fall semester where j is the SCH

level.

12| N(i+1) is the total number of incoming new

undergraduate students for the (i+1)th

semester.

With this terminology in place, the previously stated

assumptions of the models can now be described

algebraically:

1| When applying a model to a future period

from Fall (i+1) to Fall (i+2), the total number of

incoming students is assumed to be the same

as it was for the period used to build the model,

so it is assumed to have the value Ni+1. The

fraction of new students by SCH level for that

upcoming year is also assumed to be the same

as it was in the period used to train the models,

so each is assumed to be s(i+1)j ⁄ Ni+1. Therefore,

the estimated number of new students for a

particular SCH level in that future year can be

obtained by multiplying the value of this fraction

by the estimated total number of students in

the current year. That is, the estimate for the

number of new students in the future year for

that particular SCH level is given by

s(i+1)j ⁄ Ni+1 × Ni+1 = s(i+1)j.

2| The fractional loss and fractional continuation

ratios are also assumed to be fixed by SCH level.

In other words, for a future year these ratios

are assumed to be lij ⁄ hij and cij ⁄ hij, the same as

they were in the year used to build the model.

Therefore, for the upcoming future period from

Fall (i+1) to Fall (i+2), the estimated number of

lost and continuing students for the jth SCH

level are obtained by multiplying these ratios by

the number of students h(i+1)j in that SCH level in

the current Fall (i+1). This multiplication is lij ⁄ hij

× h(i+1)j to estimate lost students in the jth SCH

level and cij ⁄ hij × h(i+1)j to estimate continuing

students in the jth SCH level.

3| Finally, the fractional flow from a particular SCH

level to another SCH level is assumed to be

fixed. In other words, for a future year these

ratios are assumed to be dijk ⁄ cij, the same as

they were in the year used to build the model.

Therefore, for the upcoming future period from

Fall (i+1) to Fall (i+2), the estimated number

of students transitioning from SCH level j to

SCH level k is given by the value of this ratio

dijk ⁄ cij multiplied by the estimated number of

continuing students in the jth SCH level.

The processes described above can be applied

iteratively to obtain estimates for years even farther

into the future by using the estimated values from

one iteration as inputs into the next iteration.

Using the terms and formulae, we created a

spreadsheet matrix (Table 1) that includes the

various credit hour classifications as well as the

nonabsorbed transient student states and the

absorbed state of no longer enrolled.

30Fall 2019 Volume

From this SCH flow structure, we can observe

the relationships of credit hour flow between

and among the various states, including flow into

nonabsorbing states (staying or moving into another

credit hour state) or into absorbing states (not

enrolling at the institution). The relationships among

the variables are as follows:

1| cij = ∑ nk=1 dijk represents those current students

who were in SCH level j who stayed at the

institution.

2| oij = ∑ nk=1k≠j

dijk represents those current students

who were in SCH level j who moved to all other

SCH levels.

3| wik = ∑ nj=1j≠k

dijk represents those current students

who were in SCH levels other than k who moved

to SCH level k.

4| Hi = ∑ nj=1 hij represents semester headcount at

Fall semester i.

5| Li = ∑ nj=1 lij represents those students at Fall

semester i who did not reenroll.

6| Ci = ∑ nj=1 cij represents those students at Fall

semester i who did reenroll.

The following relationship,

∑ ∑n

k=1

n

j=1wik = oij

shows two equivalent ways of expressing the

collection of students who remain at the institution

and move from any SCH level to a different SCH

level during the year. Conservation of student flow is

obtained only when students from level j stay in SCH

Table 1. Basic Structure Matrix of the Markov Chain ModelSC

H L

evel

s

Fall

i Hea

dcou

nt

Lost

Cont

inui

ngMovement from SCH Level

Stat

ic

Infl

ow

Out

flow

Fall

i+1,

New

Stu

dent

s

Fall

i+1

Hea

dcou

nt

1 2 . . n

1 hi1 li1 ci1

Mov

emen

t to

SCH

Lev

el 1 di11 di11 wi1 oi1 s(i+1)1 h(i+1)1

2 hi2 li2 ci2 2 di12 di22 di22 wi2 oi2 s(i+1)2 h(i+1)2

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

n hin lin cin n di1n . . . dinn dinn win oin s(i+1)n h(i+1)n

Note: This table shows the basic structure matrix of the headcount SCH flow associated with the Markov chain model that connects

the undergraduate headcount in the ith Fall to the headcount in the (i+1)th Fall.

31Fall 2019 Volume

level j or move to other SCH levels, or when students

from other SCH levels move into SCH level j.

Given these relationships, the number of

undergraduates by level in the second Fall semester

can be calculated using the following formula:

h(i+1)j = hij – ij – oij + wij + s(i+1)j

This is the number of total transient students in

one of the SCH levels after 1 year who were not

absorbed by withdrawing or graduating. Therefore,

the total number of students in the (i+1)th Fall

semester is simply given by

Hi+1 = Hi – Li – Ni+1

since the inflow and outflow terms cancel upon

summation.

RESULTSThe model used actual data from a Southeastern,

masters-level (Large Programs) public institution

for Fall 2010 through Fall 2017. The enrollments for

these 8 years are displayed in Table 2.

In developing the Markov chain matrix for each year,

the total number of students within each category

were noted and tracked to the following year. Within

this matrix, one can observe the various student

states by each category to determine who is moving

into transitional (nonabsorbing) states and who is

graduating or not returning. These more-granular

data within the matrix offer clues as to when

students may be leaving the institution and where

there are potential bulges in the system coming

from new or transfer students.

Table 2. Annual Enrollment Data, Fall 2010–Fall 2017

Fall iFall i

Headcount Lost Continuing NewFall (i+1)

Headcount

Fall 2010 9,652 3,773 5,879 3,957 9,836

Fall 2011 9,836 4,082 5,754 3,721 9,475

Fall 2012 9,475 3,965 5,510 3,761 9,271

Fall 2013 9,271 3,843 5,428 3,574 9,002

Fall 2014 9,002 3,685 5,317 3,598 8,915

Fall 2015 8,915 3,792 5,123 3,993 9,116

Fall 2016 9,116 3,945 5,171 3,919 9,090

Fall 2017 9,090 not known not known not known not known

32Fall 2019 Volume

28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

SCH Level Groupings

(>162)

(157–162)

(151–156)

(145–150)

(139–144)

(133–138)

(127–132)

(121–126)

(115–120)

(109–114)

(103–108)

(97–102)

(91–96)

(85–90)

(79–84)

(73–78)

(67–72)

(61–66)

(55–60)

(49–54)

(43–48)

(37–42)

(31–36)

(25–30)

(19–24)

(13–18)

(7–12)

(0–6) SCH Level Definition

257

49 61 96

113

135

154

179

237

271

323

341

400

347

347

346

369

446

399

291

316

332

386

501

322

245

264

1589 HC1 (Fall 2016)

176

25 45 78 79 89

116

131

171

188

230

204

248

165

121

84 85

105

114

92 94

108

121

147

141

110

81

597 Lost

81 24 16 18 34 46 38 48 66 83 93

137

152

182

226

262

284

341

285

199

222

224

265

354

181

135

183

992 Continuing

4 1 4 2 0 1 1 3 5 3 5 8 11 9 13 18 16 37 31 19 16 13 17 11 1 1 0 0

GradA16

4 1 2 0 0 1 1 2 5 3 1 6 6 5 12 12 13 27 21 17 13 8 13 9 1 0 0 0

GradA16E

124

21 32 56 48 56 90 98

135

159

188

147

175

100

34 14 1 0 0 0 0 0 0 0 0 0 0 0

GradB16

2 1 0 0 1 3 2 3 3 2 5 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0GradB16E

Movement to SCH Level Number

28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1 3 15 60

258

408

158

60 14 1

Movem

ent from SCH

Level Num

ber

1 3 5 21 85 42 10 6 5 2

4 13 45 35 18 9 5 3

1 4 16 46 58 31 13 7 4

3 5 38

115

103

57 19 11 5

1 1 6 24 89 82 31 19 5 1 6

1 1 10 19 70 59 33 17 9 7

3 8 29 69 43 39 17 11 8

1 1 7 31 56 49 27 13 10 9

5 9 71 84 58 26 21 9 1 10

4 20 53

123

79 28 23 9 11

4 18 46 85 79 25 13 12 12

1 1 2 5 5 45 79 64 34 18 7 13

1 8 28 62 84 17 17 8 14

1 8 18 60 52 28 10 2 15

2 1 16 38 49 23 13 5 16

1 2 4 16 27 44 19 16 5 17

12 25 18 17 10 8 18

2 1 4 22 23 11 11 6 19

1 4 10 21 8 12 7 1 20

1 6 10 14 6 6 3 21

4 8 10 4 4 4 22

18 6 9 6 5 23

20 3 5 5 24

13 1 3 25

16 26

24 2728

81 0 0 1 1 2 4 2 2 3 3 3 5 3 1 1 2 2 1 4 3 5 6 3 5 6 5 15 Static

97 27 42 46 77 84 98

148

182

228

281

245

336

255

202

207

200

253

280

230

224

235

342

435

169

65 14 0

Inflow to

0 24 16 17 33 44 34 46 64 80 90

134

147

179

225

261

282

339

284

195

219

219

259

351

176

129

178

977 Outflow from

109

10 14 22 20 32 41 45 50 25 46 62 87 77 96

107

105

141

116

85 79

103

116

130

161

195

239

1606 New

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 113

161

195

239

1606 NewUnder30Hrs

34 4 2 5 2 12 12 17 19 7 19 21 24 25 45 56 63

103

74 38 32 45 57 60 37 46 23 32 Transfer

287

37 56 69 98

118

143

195

234

256

330

310

428

335

299

315

307

396

397

319

306

343

464

568

335

266

258

1621 HC2 (Fall 2017)

Table 3. Fall 2016 to Fall 2017 6-SCH M

atrix

33Fall 2019 Volume

Table 3 represents one such matrix, the 6-SCH

matrix from Fall 2016 to Fall 2017. The 28 6-SCH

groupings are labeled down the left with the same

corresponding 28 groupings across the center of

the matrix. This table also contains headcount by

groupings, how many within each grouping did not

return, how many graduated, and how many new

students enrolled in Fall 2017 but not Fall 2016.

Matrices such as this one can be examined to

identify the aforementioned leaks and bulges in the

enrollment pipeline.

The following labels are used in Table 3:

1| HC1 (Fall 2016): Fall 2016 census undergraduate

enrollment excluding special groups.

2| Lost: Enrolled Fall 2016 but not in Fall 2017.

This includes students that graduated without

reenrolling, as a subset. When determining

if the student returned in Fall 2017, only

undergraduate students, excluding special

groups, were considered.

3| Continuing: Enrolled in Fall 2016 and Fall 2017.

4| GradA16: Awarded an associate degree in

Fall, Spring, or Summer of Academic Year

2016–17. Note that only one degree is counted

per student to avoid double-counting, with

bachelor’s degrees given precedence over

associate’s degrees.

5| GradA16E: Awarded an associate’s degree and

enrolled in next Fall term in another degree

program. These students are a subset of

GradA16.

6| GradB16: Awarded a bachelor’s degree in Fall,

Spring, or Summer of Academic Year 2016–17.

7| GradB16E: Awarded a bachelor’s degree and

enrolled in next Fall term in another degree

program. These students are a subset of

GradB16.

8| Columns in the center indicate movement of

continuing students from the Fall 2016 SCH

categories to the Fall 2017 SCH categories. Note

that the central portion of Table 3 does not

include counts for students who enrolled both

semesters but remained in the same SCH level;

these counts are instead separately labeled

Static.

9| Static: Enrolled in Fall 2016 and Fall 2017 and

stayed in the same SCH level.

10| Inflow to: Enrolled in Fall 2016 within a different

SCH level but moved to the current SCH level in

Fall 2017.

11| Outflow from: Enrolled in the SCH level during

Fall 2016 but moved to another SCH level in Fall

2017.

12| New: Enrolled in Fall 2017 but did not enroll in

Fall 2016. (NewUnder30Hrs and Transfer are

subsets of New.)

13| NewUnder30Hrs: New students with fewer than

30 hours.

14| Transfer: Transfer students.

15| HC2: Fall 2017 census undergraduate

enrollment excluding special groups.

According to the table, in Fall 2016 there were 1,589

students in the (0–6) SCH group. Out of these, 597

did not return the next Fall semester. A total of

408 of these students transitioned into the (25–30)

SCH group, indicating that they were progressing

normally, while 232 transitioned into groups of 24

or fewer SCH. With a quick examination of the flow,

it is easy to see that the majority of students are

not returning within the SCH groupings that make

up the freshman and sophomore years as denoted

in the Lost column. In the (85–90) SCH grouping,

109 students graduated, and 5 of the students

who graduated reenrolled in Fall 2017, meaning

34Fall 2019 Volume

that 104 of the students who graduated did not

reenroll. A total of 165 students in the (85–90) SCH

grouping were lost (did not reenroll); subtracting the

aforementioned 104 students leaves 61 students

who neither graduated nor reenrolled.

A total of 914 new transfer students entered for Fall

2017, indicating a significant number of students

who took some type of transfer credit. Many of

these new transfers could constitute dual-enrolled

students who took both high school and college

classes. The bulk of the new transfer students,

however, are entering with more than 54 and fewer

than 84 SCHs.

In observing the higher groupings, the table

indicates that 865 students had accumulated more

than 126 SCH and 448 (52%) graduated. Of the

students who earned more than 126 SCHs, 608 did

not reenroll in the institution.

While this table represents only one of the six

matrices created for this study, the possibilities of

tracking student flow by groupings, classifications,

or years are numerous. Moreover, it can be argued

that the process of tracking student flow through

transitional states within the Markov process is

somewhat intuitive and indicative of the strong

predictive properties of the model.

Table 4 shows the predictions for the next 3 years,

along with the actual data. The model was built using

the flow of students over a particular academic

year. There were six such academic years used for

Table 4. Actual Enrollment and Predictions, Fall 2012–Fall 2017

Model Fall 12 Fall 13 Fall 14 Fall 15 Fall 16 Fall 17

Reality Actual Headcounts 9,475 9,271 9,002 8,915 9,116 9,090

Model 10_11 6SCH Predicted Headcounts % Diff. from Actual

9,948 4.99%

9,999 7.85%

10,002 11.11%

Model 11_12 6SCH Predicted Headcounts % Diff. from Actual

9,244 –0.29%

9,076 0.82%

8,958 0.48%

Model 12_13 6SCH Predicted Headcounts % Diff. from Actual

9,105 1.14%

8,980 0.73%

8,903 –2.34%

Model 13_14 6SCH Predicted Headcounts % Diff. from Actual

8,839 –0.85%

8,745 –4.07%

8,694 –4.36%

Model 14_15 6SCH Predicted Headcounts % Diff. from Actual

8,874 –2.65%

8,865 –2.48%

Model 15_16 6SCH Predicted Headcounts % Diff. from Actual

9,258 1.85%

Note: The model creates predictions for the next 3 years (when actual data are available for comparison) for each of the models

using the 6-SCH methods.

35Fall 2019 Volume

construction of the models. The columns of Table 4

show the years for which an enrollment prediction

was generated. As can be seen in the table,

predictions for the 10_11 Model for both methods

were overspecified by about 5% for Fall 2012

and about 11% for Fall 2014. The 11_12 Models

produced better projections, coming within less than

1% of the actual values for all 3 years. The prediction

of the 12_13 Model differed from the actual

enrollment by an average of –0.2%. Results from the

13_14 Model indicate that the prediction differed

by an average of 3.1%. In most cases, predictions

farther into the future from the years used to train

the models have greater residuals, which is to be

expected in any forecasting problem.

We calculated averages of the absolute values of

the percentage differences between the actual and

predicted values for enrollment using the actual and

predicted enrollment from Table 4. The percentage

difference between the predicted and actual value is

defined as

% difference = predicted value - actual value

actual valuex 100%

We can examine the predictive ability of the models

by using the average value of the absolute values

of these percentage differences, because these

values show on average how far off the models

were, regardless of sign. In a mathematical sense,

the absolute value between two numbers is known

as the standard Euclidean distance between two

points and indicates the real distance between two

numbers (Bartle & Sherbert, 2011). The results as

shown in Table 5 clearly indicate that the predictive

ability of the model decreases as number of

years out from the years used to build the model

increases, which is expected, similar to how weather

forecasts become less accurate the farther they go

into the future.

Based on the results from Table 5, the study will

examine only 1-year-out predictions, because these

were the most accurate. The actual values are

compared with those 1-year-out predictions in Table

6. The predicted enrollment for Fall X in Table 6 is

produced from the enrollment data from Fall X-2

and Fall X-1 and subsequent flow rates from Fall X-2

to Fall X-1.

Note that the 6-year average of the absolute

values of the percentage differences by class range

from 2.8% to 4.7%. The 2016 freshman percent

difference of –12.9% represents an outlier due

to a major university initiative to increase new

freshmen enrollment. This influx of new freshmen

was significantly different from past years and clearly

signals the bulge in the student flow pipeline as

mentioned above. By utilizing the iterative process

of producing Fall X projections from the enrollment

data from Fall X-2 and subsequent flow rates

from Fall X-2 to Fall X-1, the effect of this bulge in

the system can be tracked into the future to plan

upcoming course offerings.

Table 5. Mean Absolute Value of Percent Differences by Years Out for 6-SCH Models

Prediction Time Frame

Mean Absolute Value of Percent Difference

1 year out 1.96%

2 years out 3.19%

3 years out 4.57%

36Fall 2019 Volume

By observing the predictive capabilities of the model,

it is easy to see how administrators and enrollment

managers can use these results to plan for classes

and instructional personnel. Here, both annual

projections and classification average projections for

the 5-year period were off by no more than 6.6%,

which should fall within the margin of error for most

larger institutions.

Furthermore, Monte Carlo simulation could be

used to obtain enrollment predictions that give a

range of plausible values instead of a single point

estimate for a future year’s enrollment. Monte Carlo

simulations have been used in the context of higher

education by Torres, Crichigno, and Sanchez (2018)

to examine degree plans for potential bottlenecks.

In applying these methods to this enrollment model,

the fractions of students transitioning between

specific levels would be treated more like the result

of many coin flips than as fixed fractional values, and

the ranges of predicted values could be obtained by

repeated random simulation. This level of simulation

was not performed in this study.

Table 6. The 6-SCH Models’ 1-Year-Out Predictions Compared to Actual Enrollment, 2012–17

Freshman Sophomore Juniors Seniors All Levels

Mean Absolute % Difference of

Class Levels

2012 Actual Predicted % Difference

2,876 3,114 8.28%

2,035 2,090 2.68%

1,871 1,966 5.10%

2,693 2,778 3.17%

9,475 9,948 5.00% 4.81%

2013 Actual Predicted % Difference

2,729 2,817 3.23%

1,890 1,875

–0.79%

1,870 1,834

–1.92%

2,782 2,718

–2.30%

9,271 9,244

–0.29% 2.06%

2014 Actual Predicted % Difference

2,644 2,709 2.47%

1,803 1,800

–0.16%

1,870 1,789

–4.36%

2,685 2,807 4.53%

9,002 9,105 1.14%

2.88%

2015 Actual Predicted % Difference

2,533 2,574 1.60%

1,944 1,809

–6.93%

1,738 1,816 4.51%

2,700 2,640

–2.24%

8,915 8,839

–0.85% 3.82%

2016 Actual Predicted % Difference

2,921 2,543

–12.93%

1,724 1,885 9.36%

1,855 1,801

–2.89%

2,616 2,644 1.07%

9,116 8,874

–2.65% 6.56%

2017 Actual Predicted % Difference

3,048 3,053 0.17%

1,829 1,788

–2.24%

1,652 1,766 6.91%

2,561 2,651 3.51%

9,090 9,258 1.85% 3.21%

Mean Absolute % Difference

4.78% 3.69% 4.28% 2.80% 1.96%

37Fall 2019 Volume

CONCLUSIONSThe use of Markov chains in projecting enrollment

and the management thereof has gained popularity

among professionals in higher education. The

short-term projections created by this stochastic

process are unique to other time-tested forecasting

tools used in enrollment management. When

used properly, Markov chains can aid institutions

in determining progression of students that

are different from more-traditional ARIMA and

regression prediction tools in that:

1| they can give accurate enrollment predictions

with only 2 previous years’ data, which can be

helpful when large longitudinal databases are

not available;

2| they can be used to generate predictions on

segments of a group of students rather than the

entire population, which may be required for

other models; and

3| the almost intuitive nature of the Markov

chain lends well to changes in student flow

characteristics, which often cannot be explained

by a complex statistical formula.

By creating groupings and tracking students within

those groupings by the state they transition into, the

researcher can also get a better picture of what type

of students are leaving and when they are leaving.

As shown in this study, the strong predictability

of Markov chains allows administrators to better

plan course scheduling and instructor demand

while managing tight budgets. In this study, several

predictive headcount models were developed using

SCH flow as the annual driver. Eight years of Fall

enrollment data from the institution were used to

develop the models. When applied to historical

data each gives 1-year-out predictions within a

calculated level of uncertainty. The models can easily

be modified to change the new student input data,

the continuation rates, and the interlevel flow rates,

should that be desired. Furthermore, similar models

could be used to track Fall to Spring retention as well

as Spring to Fall retention.

REFERENCESBartle, R. G., & Sherbert, D. R. (2011). Introduction

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Borden, V. M. H., & Dalphin J. F. (1998). Simulating

the effect of student profile changes on retention

and graduation rates: A Markov chain analysis. Paper

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Brezavšček, A., Bach, M., & Baggia, A. (2017). Markov

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progress in higher education. Organizacija, 50(2),

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Clagett, C. A. (1991). Institutional research: The key

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Coomes, M. D. (2000). The historical roots of

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issues, and methods: New directions for institutional

research, 93 (51–65). San Francisco: Jossey-Bass.

38Fall 2019 Volume

Ewell, P. T. (1985). Recruitment, retention and

student flow: A comprehensive approach to

enrollment management. National Center for Higher

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Boulder, CO.

Fallows, J., & Ganeshananthan, V. (2004, October).

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picture/303520/

Gagne, L. (2015). Modeling the progress and

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Hossler, D. (1984). Enrollment management: An

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management: A historical perspective. Journal of

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Liljegren, A., & Saks, M. (Eds.). (2017). Professions and

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Oliver, R. M. (1968). Models for predicting gross

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Torraco, R. J., & Hamilton, D. W. (2013). The leaking

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39Fall 2019 Volume

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